Let a,b,c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that $x = c y + b z, y = a z + c x$ , and $z = b x + a y$ .Then $a^{2} + b^{2} + c^{2} + 2 a b c$ is equal to

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

No worries! We‘ve got your back. Try BYJU‘S free classes today!

-1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A 1
Given, $x = c y + b z$
$y = a z + c x \Rightarrow y = a z + c (c y + b z) = a z + b c z + c^{2} y$
$y (1 - c^{2}) = a z + b c z$
$y = \frac{(a + b c) z}{(1 - c^{2})}$
$x = \frac{c (a + b c) z}{(1 - c^{2})} + b z$
$x = \frac{a c z + b c^{2} z + b z - b c^{2} z}{(1 - c^{2})}$
$= \frac{(a c + b) z}{1 - c^{2}}$
$z = b x + a y$
$z = \frac{b z (a c + b)}{1 - c^{2}} + \frac{a z (a + b c)}{1 - c^{2}}$
$1 - c^{2} = a b c + b^{2} + a^{2} + a b c$
$1 = a^{2} + b^{2} + c^{2} + 2 a b c$

Suggest Corrections

Similar questions

Q. Let

a . b, c

be any real numbers, suppose that there are real number

x, y, z

not all zero such that

x = c y + b z, y = a z + c x, z = b x + a y,

then

a^{2} + b^{2} + c^{2} + 2 a b c

equal to

Let a,b,c,be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x= cy +bz, y=az + cx and z = bx +ay. Then , a² + b²+ c²+ 2abc is equal to

Q. Let a,b,c be any real numbers. Suppose that there are real numbers

x, y, z

not all zero such that

x = c y + b z, y = a z + c x

and

z = b x + a y .

Then

a^{2} + b^{2} + c^{2} + 2 a b c

is equal to

Q. Let a,b and c be any real numbers. Suppose that there are real numbers

x, y, z

not all zero such that

x = c y

y = a z + c x

and

z = b x + a y

, then

a^{2} + b^{2} + c^{2} + 2 a b c

is equal to:

Q. Given,

x = c y + b z, y = a z + c x, z = b x + a y

, where x, y, z are not all zero, and a, b, c are real numbers. The value of

a^{2} + b^{2} + c^{2} + 2 a b c

Join BYJU'S Learning Program

Let a,b,c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x=cy+bz, y=az+cx, and z=bx+ay.Then a2+b2+c2+2abc is equal to

The correct option is A 1Given, x=cy+bzy=az+cx⇒y=az+c(cy+bz)=az+bcz+c2yy(1−c2)=az+bczy=(a+bc)z(1−c2)x=c(a+bc)z(1−c2)+bzx=acz+bc2z+bz−bc2z(1−c2)=(ac+b)z1−c2z=bx+ayz=bz(ac+b)1−c2+az(a+bc)1−c21−c2=abc+b2+a2+abc1=a2+b2+c2+2abc

Let a,b,c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that $x = c y + b z, y = a z + c x$ , and $z = b x + a y$ .Then $a^{2} + b^{2} + c^{2} + 2 a b c$ is equal to